IDEAS home Printed from https://ideas.repec.org/p/arx/papers/0710.5301.html
   My bibliography  Save this paper

An iterative algorithm for evaluating approximations to the optimal exercise boundary for a nonlinear Black-Scholes equation

Author

Listed:
  • Daniel Sevcovic

Abstract

The purpose of this paper is to analyze and compute the early exercise boundary for a class of nonlinear Black--Scholes equations with a nonlinear volatility which can be a function of the second derivative of the option price itself. A motivation for studying the nonlinear Black--Scholes equation with a nonlinear volatility arises from option pricing models taking into account e.g. nontrivial transaction costs, investor's preferences, feedback and illiquid markets effects and risk from a volatile (unprotected) portfolio. We present a new method how to transform the free boundary problem for the early exercise boundary position into a solution of a time depending nonlinear parabolic equation defined on a fixed domain. We furthermore propose an iterative numerical scheme that can be used to find an approximation of the free boundary. We present results of numerical approximation of the early exercise boundary for various types of nonlinear Black--Scholes equations and we discuss dependence of the free boundary on various model parameters.

Suggested Citation

  • Daniel Sevcovic, 2007. "An iterative algorithm for evaluating approximations to the optimal exercise boundary for a nonlinear Black-Scholes equation," Papers 0710.5301, arXiv.org.
    • Handle: RePEc:arx:papers:0710.5301
    as

    Download full text from publisher

    File URL: http://arxiv.org/pdf/0710.5301
    File Function: Latest version
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Leland, Hayne E, 1985. "Option Pricing and Replication with Transactions Costs," Journal of Finance, American Finance Association, vol. 40(5), pages 1283-1301, December.
    2. Roll, Richard, 1977. "An analytic valuation formula for unprotected American call options on stocks with known dividends," Journal of Financial Economics, Elsevier, vol. 5(2), pages 251-258, November.
    3. Rachel Kuske & Joseph Keller, 1998. "Optimal exercise boundary for an American put option," Applied Mathematical Finance, Taylor & Francis Journals, vol. 5(2), pages 107-116.
    4. Avellaneda Marco & ParaS Antonio, 1994. "Dynamic hedging portfolios for derivative securities in the presence of large transaction costs," Applied Mathematical Finance, Taylor & Francis Journals, vol. 1(2), pages 165-194.
    5. Geske, Robert & Johnson, Herb E, 1984. "The American Put Option Valued Analytically," Journal of Finance, American Finance Association, vol. 39(5), pages 1511-1524, December.
    6. Rüdiger Frey & Alexander Stremme, 1997. "Market Volatility and Feedback Effects from Dynamic Hedging," Mathematical Finance, Wiley Blackwell, vol. 7(4), pages 351-374, October.
    7. Black, Fischer & Scholes, Myron S, 1973. "The Pricing of Options and Corporate Liabilities," Journal of Political Economy, University of Chicago Press, vol. 81(3), pages 637-654, May-June.
    8. Halil Mete Soner & Guy Barles, 1998. "Option pricing with transaction costs and a nonlinear Black-Scholes equation," Finance and Stochastics, Springer, vol. 2(4), pages 369-397.
    9. Geske, Robert & Roll, Richard, 1984. "On Valuing American Call Options with the Black-Scholes European Formula," Journal of Finance, American Finance Association, vol. 39(2), pages 443-455, June.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Zhongkai Liu & Tao Pang, 2016. "An efficient grid lattice algorithm for pricing American-style options," International Journal of Financial Markets and Derivatives, Inderscience Enterprises Ltd, vol. 5(1), pages 36-55.
    2. Chinonso Nwankwo & Weizhong Dai, 2020. "Explicit RKF-Compact Scheme for Pricing Regime Switching American Options with Varying Time Step," Papers 2012.09820, arXiv.org, revised Feb 2022.
    3. Chinonso I. Nwankwo & Weizhong Dai & Ruihua Liu, 2023. "Compact Finite Difference Scheme with Hermite Interpolation for Pricing American Put Options Based on Regime Switching Model," Computational Economics, Springer;Society for Computational Economics, vol. 62(3), pages 817-854, October.
    4. Tomas Bokes & Daniel Sevcovic, 2009. "Early exercise boundary for American type of floating strike Asian option and its numerical approximation," Papers 0912.1321, arXiv.org.
    5. Chinonso I. Nwankwo & Weizhong Dai, 2024. "Efficient adaptive strategies with fourth-order compact scheme for a fixed-free boundary regime-switching model," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 47(1), pages 43-82, June.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Lim, Terence & Lo, Andrew W. & Merton, Robert C. & Scholes, Myron S., 2006. "The Derivatives Sourcebook," Foundations and Trends(R) in Finance, now publishers, vol. 1(5–6), pages 365-572, April.
    2. Weihan Li & Jin E. Zhang & Xinfeng Ruan & Pakorn Aschakulporn, 2024. "An empirical study on the early exercise premium of American options: Evidence from OEX and XEO options," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 44(7), pages 1117-1153, July.
    3. Maria do Rosário Grossinho & Yaser Faghan Kord & Daniel Sevcovic, 2017. "Pricing American Call Option by the Black-Scholes Equation with a Nonlinear Volatility Function," Working Papers REM 2017/18, ISEG - Lisbon School of Economics and Management, REM, Universidade de Lisboa.
    4. Maria do Rosario Grossinho & Yaser Faghan Kord & Daniel Sevcovic, 2017. "Analytical and numerical results for American style of perpetual put options through transformation into nonlinear stationary Black-Scholes equations," Papers 1707.00356, arXiv.org.
    5. Maria do Rosario Grossinho & Yaser Kord Faghan & Daniel Sevcovic, 2016. "Pricing Perpetual Put Options by the Black-Scholes Equation with a Nonlinear Volatility Function," Papers 1611.00885, arXiv.org, revised Nov 2017.
    6. Maria do Rosário Grossinho & Yaser Kord Faghan & Daniel Ševčovič, 2017. "Pricing Perpetual Put Options by the Black–Scholes Equation with a Nonlinear Volatility Function," Asia-Pacific Financial Markets, Springer;Japanese Association of Financial Economics and Engineering, vol. 24(4), pages 291-308, December.
    7. Karol Duris & Shih-Hau Tan & Choi-Hong Lai & Daniel Sevcovic, 2015. "Comparison of the analytical approximation formula and Newton's method for solving a class of nonlinear Black-Scholes parabolic equations," Papers 1511.05661, arXiv.org, revised Nov 2015.
    8. Daniel Sevcovic & Magdalena Zitnanska, 2016. "Analysis of the nonlinear option pricing model under variable transaction costs," Papers 1603.03874, arXiv.org.
    9. M. Rezaei & A. R. Yazdanian & A. Ashrafi & S. M. Mahmoudi, 2022. "Numerically Pricing Nonlinear Time-Fractional Black–Scholes Equation with Time-Dependent Parameters Under Transaction Costs," Computational Economics, Springer;Society for Computational Economics, vol. 60(1), pages 243-280, June.
    10. Wei, Dongming & Erlangga, Yogi Ahmad & Zhumakhanova, Gulzat, 2024. "A finite element approach to the numerical solutions of Leland’s model," International Review of Economics & Finance, Elsevier, vol. 89(PA), pages 582-593.
    11. Maria do Rosário Grossinho & Yaser Faghan Kord & Daniel Sevcovic, 2017. "Pricing Perpetual Put Options by the Black-Scholes Equation with a Nonlinear Volatility Function," Working Papers REM 2017/19, ISEG - Lisbon School of Economics and Management, REM, Universidade de Lisboa.
    12. Maria do Rosario Grossinho & Yaser Faghan Kord & Daniel Sevcovic, 2017. "Pricing American Call Options by the Black-Scholes Equation with a Nonlinear Volatility Function," Papers 1707.00358, arXiv.org, revised Jun 2018.
    13. Bjork, Tomas, 2009. "Arbitrage Theory in Continuous Time," OUP Catalogue, Oxford University Press, edition 3, number 9780199574742.
    14. Al–Zhour, Zeyad & Barfeie, Mahdiar & Soleymani, Fazlollah & Tohidi, Emran, 2019. "A computational method to price with transaction costs under the nonlinear Black–Scholes model," Chaos, Solitons & Fractals, Elsevier, vol. 127(C), pages 291-301.
    15. Cristina Viegas & José Azevedo-Pereira, 2020. "A Quasi-Closed-Form Solution for the Valuation of American Put Options," IJFS, MDPI, vol. 8(4), pages 1-16, October.
    16. Suresh M. Sundaresan, 2000. "Continuous‐Time Methods in Finance: A Review and an Assessment," Journal of Finance, American Finance Association, vol. 55(4), pages 1569-1622, August.
    17. Cosma, Antonio & Galluccio, Stefano & Scaillet, Olivier, 2012. "Valuing American options using fast recursive projections," Working Papers unige:41856, University of Geneva, Geneva School of Economics and Management.
    18. Olivier Guéant, 2016. "The Financial Mathematics of Market Liquidity: From Optimal Execution to Market Making," Post-Print hal-01393136, HAL.
    19. Perrakis, Stylianos & Lefoll, Jean, 2000. "Option pricing and replication with transaction costs and dividends," Journal of Economic Dynamics and Control, Elsevier, vol. 24(11-12), pages 1527-1561, October.
    20. Joel Vanden, 2006. "Exact Superreplication Strategies for a Class of Derivative Assets," Applied Mathematical Finance, Taylor & Francis Journals, vol. 13(1), pages 61-87.

    More about this item

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:arx:papers:0710.5301. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: arXiv administrators (email available below). General contact details of provider: http://arxiv.org/ .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.